Problem 1 - 20 pts.

Suppose I have a discrete state variable model, , and I use the similarity transformation, x(kT) = Pz(kT), to obtain the following decoupled equivalent discrete state variable model:

a) Explain how I find the similarity transformation matrix P.

b) Draw a block diagram for the decoupled system

c) What are the controllable eigenvalues? What are the observable eigenvalues? What are the stable eigenvalues?

d) What is the transfer function, H(z)=Y(z)/W(z)

e) Suppose the decoupled initial state is z(0T)=[3 0]T. Using the method of your choice, find z(kT) if w(kT) is a discrete unit step.

f) Now find x(kT) if our similarity transformation matrix is

g) Lastly, use your knowledge of linearity and shift invariance to find x(kT) if z(2T)=[9 0]T and the input is w(kT)=5u((k-1)T).

Problem 2 - 20 pts.

Suppose we are given the following impulse response of a linear, shift-invariant system:

a) We learned in class that the output of such a system is given by the convolution sum, . If our input is w(kT)=4[u(kT)-u((k-2)T)], use the axis provided and graphical convolution to calculate y(4T) (just do y(4T)).

b) Now use analytical convolution (i.e., ) to find an expression for y(kT) that is valid for all k

c) Next, check your answer to part b) by using Z-transforms

d) Finally, find the response of the system due to the input w(kT)=8[u((k-3)T)-u((k-5)T)]

Problem 5 - 20 pts.

a) Recall that is the solution to the zero-input discrete state variable model,. Use this information to derive the Z-transform of

b) Prove the convolution property of the Z-transform

c) Next, use the convolution property of the Z-transform to derive the solution to the entire next state equation,